## number theory

For students of class 9-10 (age 14-16)
Mahfuz Sobhan
Posts: 25
Joined: Sat Feb 07, 2015 5:40 pm

### number theory

Let us define $$S_n$$ to be the set of all integers divisible by $$2014^n$$ but not $$2014^n+^1$$ where $$n$$ is a non-negative integer. What is the value of $$n$$ (if any) so that $$500!$$ belongs to $$S_n$$.

seemanta001
Posts: 13
Joined: Sat Jun 06, 2015 9:31 am
Location: Chittagong

### Re: number theory

We can observe that $$2014=2\times19\times53$$.Here $53$ is the biggest prime factor of $2014$.
We have to find $\sum\frac{500}{53^n}$ for $n>0$.
There are $9$ numbers that are multiples of $53$ from $1$ to $500$.
No other multiples of $53^n$ are within numbers $1$ to $500$,where $n>1$.